Problem: What is the value of $\dfrac{d}{dx}\left(x^{^{\scriptsize\dfrac{1}{3}}}\right)$ at $x=8$ ?
Explanation: Let's first find the expression for $\dfrac{d}{dx}\left(x^{^{\frac{1}{3}}}\right)$ and then evaluate it at $x=8$. The derivative can be found using the power rule : $\dfrac{d}{dx}(x^n)=n\cdot x^{n-1}$ (Remember that this applies even when $n$ is a fraction.) $\begin{aligned} &\phantom{=}\dfrac{d}{dx}\left(x^{^{\frac{1}{3}}}\right) \\\\ &=\dfrac{1}{3}x^{^{\frac{1}{3}-1}} \gray{\text{The power rule}} \\\\ &=\dfrac13x^{^{-\frac{2}{3}}} \end{aligned}$ So we found that $\dfrac{d}{dx}\left(x^{^{\frac{1}{3}}}\right)=\dfrac13x^{^{-\frac{2}{3}}}$, which can also be written as $\dfrac{1}{3(\sqrt[3]{x})^2}$. Now let's plug ${x=8}$ : $\begin{aligned} \dfrac{1}{3(\sqrt[3]{8})^2}&=\dfrac{1}{3(2)^2} \\\\ &=\dfrac{1}{3\cdot 4} \\\\ &=\dfrac{1}{12} \end{aligned}$ In conclusion, the value of $\dfrac{d}{dx}\left(x^{^{\frac{1}{3}}}\right)$ at $x=8$ is $\dfrac{1}{12}$.